Lectures on exceptional Lie groups:
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| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Chicago [u.a.]
Univ. of Chicago Press
1996
|
| Series: | Chicago lectures in mathematics series
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 122 S. |
| ISBN: | 0226005267 0226005275 |
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MARC
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| 100 | 1 | |a Adams, John F. |e Verfasser |4 aut | |
| 245 | 1 | 0 | |a Lectures on exceptional Lie groups |c J. F. Adams |
| 264 | 1 | |a Chicago [u.a.] |b Univ. of Chicago Press |c 1996 | |
| 300 | |a XIV, 122 S. | ||
| 336 | |b txt |2 rdacontent | ||
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Record in the Search Index
| _version_ | 1804126137100533760 |
|---|---|
| adam_text | Contents/Summary
Summary of Constructions
χ
Foreword
xi
Acknowledgments
xii
Introduction
xiii
Chapter
1.
Definitions, examples and matrix groups
1
•
Infinitesimal methods
2
Determination of tangent spaces (Lie algebras)
Adjoint representation
•
Representation theory of compact groups
3
Representations, real and symplectic
•
Weights and characters
4
«
Sketch of classification of compact Lie groups
5
The Weyl group
W
Roots
The
Stiefel
diagram
Weyl chambers, positive and simple roots
Dynkin diagrams
Reductive Lie algebras
Chapter
2.
Clifford algebras
13
Definition and universal property
Tensor products, bases for
ClţV)
•
Structure maps on Clifford algebras
15
Chapter
3.
The Spin groups I7
Definition of Pin{V) and
Spin(V)
Maximal tori
Chapter
4.
Clifford modules and representations
21
Semi-simplicity of Cl(V) and Cl(V)0
The irreducible representations
Δ, Δ+, Δ~
The weights of
Δ
31
31
Δ
is self dual.
Δ+, Δ
are self dual
(resp.
dual to each other) for
n
even
(resp. n
odd).
Behaviour of
Δ, Δ+, Δ
under restrictions
Calculations of
σ2(Δ±), λ2(Δ±)
•
The theorem of Weyl on R{G)
28
For
G
compact, connected, simply-connected R(G) is a polynomial
algebra with generators in
1-1
correspondence with the nodes of
the Dynkin diagram.
Chapter
5.
Applications of Spin representations
•
Construction of
Gì
Spin(6)
S
51/(4),
Spin(5)
s Sp(2).
Spin(5) acts transitively on
S7 C
Δ.
Sptn(6) acts transitively on 5s
x S7 C
λ1 χ Δ+.
Spin(7) acts transitively on
(i, y, z)
Є
S6 x S6 x S7,
where
x L y.
Gì
is the subgroup of Spin(7) fixing a point
z
Є
S7 C
Δ.
Gì
is compact, connected, simply connected, of rank
2,
dimension
14
acting transitively on (x,y)
Є 5е
x S6
where
íly.
•
Spin(S)
and triality
33
Ouí(5pin(8)) S
Σ3,
acting on
Δ4 , Δ , λ1.
There is a non-zero trilinear
ƒ :
Δ+
®
Δ~
®
λ1
—»
С
invariant under
Spin(8) and for real representations (of dimension
8), ƒ
takes
values in
[—1,1].
Chapter
6.
The exceptional groups: construction of Eg
37
There are pairs of Lie groups
Η
С
G
with rank
H
=
rank
G as
follows
:
local type of
H
dim
L(G)/L{H) as
С
rep.
dim
G
dim
Spin{9)
36
Δ
16
F4
52
Spm(10)xŁ/(l)/Z4
46
Δ+®ξ3+Δ-®ξ-3
32
ße
78
5pm(12)xSp(l)/Zj
69
Δ+®λ»
64
E7
133
5pin(16)/Z2
120
Δ*
128
Es
248
Construction
of
a Lie
algebra of
type £g
Standard
operating procedure
The Killing form
38
40
42
Chapter
7.
Construction of a Lie group of type Eg
45
Step
1.
A
=
L(Spin(16))
φ Δ+
is a Lie algebra with invariant inner
product and non-singular Killing form.
Step
2.
Take the group of Lie algebra automorphisms.
L{Aut(A))
=
Der(A).
If A has non-singular Killing form, then Der(A)
=
A.
Step
3.
Take the identity component.
•
Real forms of Eg
46
Chapter
8.
The construction of Lie groups of type Ft,
Ев,
Er 49
Er =
C%(SU(2)).
F4
=
C%(G2), where SU(2)
С
SU(3)
С
G2.
•
Identification of the subgroups
Η
50
•
Identification of L(G) and L(G)/L(H)
51
•
Real forms of Eg, continued
53
Chapter
9.
The Dynkin diagrams of F4,
Ев,
Er,
Eg
55
•
F4
55
•
Eg
56
•
E7
x
SU(2)/Z2
56
•
Ев
x
SU(3)
57
Chapter
10.
The Weyl group of Eg
59
W
(Eg) is the group of all automorphisms of the root system of Eg
(Theorem
10.1).
W(Eg) acts transitively on the figures (Theorem
10.13)
Chapter
11.
Representations of
Ев,
Er 69
Er
has a subgroup 5J7(8)/Zj (Theorem
11.1).
Er
has an irreducible representation of dimension
56
(Corollary
11.2).
Ев
has a subgroup SU(3)3/Z3 (Theorem
11.4).
Ев
has a 27-dimensional irreducible representation (Corollary
11.5).
Chapter
12.
Direct construction of £V 73
VIH
•
Construction
of L(Er) and its 56-dimensional representation
74
Step
1.
(Theorem
12.1)
A
=
L(SU(8)) ®
λ4
is the Lie algebra
(^ L(E7)) of maps
W -* W.
Steps
2
and
3
as for Eg.
W
admits
a skew-symmetric linear form
( , ),
a map
і
:
A
—♦
W
®
W
into symmetric tensors,
a map
о
:
W
®
W
—»
Α.
The action of E7 on
W
is faithful.
•
Et
as
&
group of maps of
W
(Cartan s construction)
80
W
admits an invariant symmetric map
W®4
—»
С.
2?7
=
the identity component of the group of linear maps
W
—►
W
preserving
( , )
and We4
-»
С
•
Real forms of Eh
82
Chapter
13.
Direct treatment of E6
85
•
Construction of Eg and its 27-dimensional representation
85
A
=
L{SU(3)3)
φ
(V! ®
Vi
® V3) is the lie algebra (^ L(£«))
of maps
W — W.
There is a non-singular inner product
( , )
on A and the Killing form
is 24(
, )
(Lemma
13.4).
There is a unique map
W
® W*
—»
A such that
(α,
iu
® w*)
=
([a,w],w ) (Lemma
13.5).
The action of
Ее
on its 27-dimensional representation is faithful.
•
Ев
as a group of maps
91
Es
is the group preserving the products
W
®
W
—►
W*,
*
W.
Chapter
14.
Direct treatment of F4,1
93
F4 has a subgroup of local type Sl/(3)
x Sl/(3)
so that the roots of
the first factor are the short roots of F4 (Lemma
14.1).
W(F4)/W(Spin(8))
S
Σ«
S Ouí(5pin(8))
(Theorem
14.2).
F4 has an irreducible 26-dimensional representation
U
(Lemma
14.4).
L(Ee)
=
L{Ft)
θ
í/,
W Ft = l + U (Lemma
14.4).
•
Structure maps on
U
96
U
carries the following structure
:
a symmetric, bilinear, non-singular, F4-invariant
6 :
U
®
U
—»
С,
a symmetric bilinear F4 invariant
ері
x
:
U
®
U
—>
U.
•
The algebra structure on
U
100
1 + U may be constructed as R3 ® Vj ® V2
Θ
V$ where {VbVi,Vi} =
{Δ+,Δ-,λ1}
(Theorem
14.18).
Chapter
15.
The Cayley numbers
105
There is a unique normed algebra of dimension
8
over
R
(Theorem
15.1).
If
Л
is a normed real algebra of dimension
8
with
1
and
1,
t, j,
ij,
£
are
orthonormal, H
=
{l,i,j,ij),
then
Η φ Η
—♦
A is an
isomorphism (Theorem
15.8).
О
=
H
©
H
is
alternative
(Theorem
15.11).
•
Connection
between the
Cayley
numbers and lie groups 111
By choosing (xo,yo,zo)
Є
Δ+ ® Δ~ ® λ1
we can identify
Δ+, Δ ,
λ1
with
О
(Theorem
15.14).
G2
=
j4uír(O)
(Theorem
15.16).
Chapter
16.
Direct treatment of F4, II: Jordan algebras
113
•
Definition and properties of the exceptional Jordan algebra
J
113
J
carries the following structure
:
a product
о
:
J ® J
—>
J,
a bilinear
6 :
J ® J
—►
R,
a linear
t
:
J
—»
R.
If we identify
R3
with the space of diagonal matrices,
Vi
with the
Cayley numbers, then R3
θ
Vi
® V2
Φ
V3 is identified with
J
so
as to make the structures correspond (Theorem
16.1).
U
corresponds to the subspace
t
= 0
in
J
and Ft acts on
J
so as
to preserve the algebraic structure (Corollary
16.2).
The algebra of polynomial functions on
J
invariant under Aut(J)
=
algebra of polynomial functions on
J
invariant under F4 (Theo¬
rem
16.6).
Fi
=
Auł(J).
The subgroup of Avt(J) fixing et
= (1,0,0)
is Spin(9).
The subgroup of Aut(J) fixing
ε1η
eit e3 is 5pin(8) (Theorem
16.7).
•
The Cayley
projective
plane
118
Appendix. Jordan algebras
119
References
121
|
| any_adam_object | 1 |
| author | Adams, John F. |
| author_facet | Adams, John F. |
| author_role | aut |
| author_sort | Adams, John F. |
| author_variant | j f a jf jfa |
| building | Verbundindex |
| bvnumber | BV011609163 |
| classification_rvk | SK 280 SK 340 |
| ctrlnum | (OCoLC)832736574 (DE-599)BVBBV011609163 |
| dewey-full | 512.55 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 512 - Algebra |
| dewey-raw | 512.55 |
| dewey-search | 512.55 |
| dewey-sort | 3512.55 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV011609163 |
| illustrated | Not Illustrated |
| indexdate | 2024-07-09T18:12:40Z |
| institution | BVB |
| isbn | 0226005267 0226005275 |
| language | English |
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| physical | XIV, 122 S. |
| publishDate | 1996 |
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| publisher | Univ. of Chicago Press |
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| series2 | Chicago lectures in mathematics series |
| spelling | Adams, John F. Verfasser aut Lectures on exceptional Lie groups J. F. Adams Chicago [u.a.] Univ. of Chicago Press 1996 XIV, 122 S. txt rdacontent n rdamedia nc rdacarrier Chicago lectures in mathematics series Ausnahmegruppe (DE-588)4221271-6 gnd rswk-swf Ausnahmegruppe (DE-588)4221271-6 s DE-604 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007819350&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Adams, John F. Lectures on exceptional Lie groups Ausnahmegruppe (DE-588)4221271-6 gnd |
| subject_GND | (DE-588)4221271-6 |
| title | Lectures on exceptional Lie groups |
| title_auth | Lectures on exceptional Lie groups |
| title_exact_search | Lectures on exceptional Lie groups |
| title_full | Lectures on exceptional Lie groups J. F. Adams |
| title_fullStr | Lectures on exceptional Lie groups J. F. Adams |
| title_full_unstemmed | Lectures on exceptional Lie groups J. F. Adams |
| title_short | Lectures on exceptional Lie groups |
| title_sort | lectures on exceptional lie groups |
| topic | Ausnahmegruppe (DE-588)4221271-6 gnd |
| topic_facet | Ausnahmegruppe |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=007819350&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT adamsjohnf lecturesonexceptionalliegroups |